"Where the fuck did you lot come from?" inquires Richard Hawley, glaring out at the crowd. This is South Yorkshire parlance for: "Hello, my fanbase – one pleasantly swollen since the release of my latest album, Standing at the Sky's Edge, which has recently been nominated for the Mercury music prize." The spotlights glint off Hawley's 1950s spectacles, but are soon sucked into the dark-matter vortex of his black leather jacket, black jeans and Brylcreemed black quiff.

Best known now for his six solo albums of crooned love songs, the Sheffield guitarist has a reputation as a hopeless romantic. But Hawley's primary interface is that of a cantankerous git hustled out of a pub before closing time. It's not the first time Hawley has played at Brixton Academy – later he'll recount the many bands he's supported here. This is, though, the first time he has headlined this major London rock staging post solo, and sold it out.

He effs, and he blinds, and he lectures his audience with the mordant wit ingested by many Sheffield musicians, along with the broken biscuits. Hawley's last job was playing guitar in Pulp: the intra-band repartee alone should probably have its own blue plaque. "How's the leg?" asks a fan, amiably. Hawley broke it early last summer, misguidedly wearing leather shoes on a marble staircase in Barcelona, while not entirely sober. He had to play a number of gigs from a wheelchair. "Nosy cunt," Hawley rejoins. The cabinet reshuffle? It was like "shitting on a shit, to pretend it's not actually a shit. And they think we can't tell!"

As with Jarvis Cocker, you could just listen to Hawley prattle on for hours. Accordingly, Radio 2 gave him his own rockabilly show. Tonight, though, at the end of a sold-out UK tour, Hawley's playing is just as eloquent.

The set opens with the title track of his most recent album, a slow, rumbly, tidal number whose gradual crescendo sets the tone for much of the evening's sculpted set. The song tells of lives blighted by violence and circumstance; Skye Edge was a notorious estate overlooking Sheffield. You might not have thought that Hawley, a loved-up 50s throwback, had much in common with rapper Plan B, whose sink-estate saga, Ill Manors, was joint favourite with Hawley to win the Mercury at the time of the shortlist announcement. But you'd be wrong.

Standing at the Sky's Edge differs markedly from most Hawley albums, replacing the bejewelled, old-time romance with brooding psychedelic rock. It is, perhaps, an iteration of psychedelia that has more in common with Oasis's Be Here Now or latterday Verve than, say, the gnarly sonic adventures of the 13th Floor Elevators. But live, the derangement reverberates more convincingly than on record.

His accomplished band thrum with an ideal admixture of discipline and sullenness, while two new backing vocalists engaged especially for tonight's gig add poignancy to Hawley's growl. On the swinging Down in the Woods, there's even an echo, perhaps, of the old Spiritualized, before Jason Pierce forsook the pursuit of the sublime for a mass audience. Drummer Dean Beresford, in particular, plays with great restraint and lucidity, never once giving in to the temptation to just thwack away on the noisier passages. Hawley himself ekes out little filigree solos with flutters of his guitar's tremolo arm, in sophisticated counterpoint to the ringing din at bass level.

It's not all swirling music for cliff edges. Hawley's older songs, such as Open Up the Door and Tonight the Streets Are Ours, are still rich with emotion, elegantly deployed. He returns for an encore with an offer to play a song he says has only aired twice before, Water Boy. First, though, there's the command to shut up and listen to the context – how he'd been taught the song by his grandfather, who had heard it performed by the remarkable American singer and committed communist Paul Robeson at a benefit for striking steelworkers in the 1920s. The water boy's job was to fetch refreshment for the slaves toiling in the field; Hawley's grandfather's own job was bucket boy, fetching beer for the steelworkers in the mill. Hawley's version is a mesmerising shard of refracted history, sweetened by a suitably liquid lap steel accompaniment from guitarist Shez Sheridan. Even better, when a heckler shouts out "We love you, Richard!" at the start, Hawley doesn't insult him. "Love you too, brother," he says, with feeling.